M.Sc Student Persovsky Diego The Natural Factor Method in Structural Analysis Department of Civil and Environmental Engineering Professor Moshe Eisenberger

Abstract

The Natural Factor Method is a matrix formulation of the structural element properties. This formulation  is parallel to the Stiffness Method, but entirely compatible with it, and there exists a simple algebraic relation between the two methods.

Deep understanding of  the element properties, both physical and mathematical, led to the Natural Factor formulation. The term “natural” is derived from the fact that the formulation is based only on the deformation of the element, by separation of the terms which represent the “Rigid Body Motion”, since they do not cause any internal force in the structural member.  There are more than one way to formulate the Natural Factor. In this research we propose two ways: The first one is by using the Natural Stiffness Matrix of an element. The Natural Stiffness matrix is formulated like an ordinary stiffness matrix, but its shape functions represent only deformations, excluding the rigid body motion shapes. The second way is the  “Eigenvalues Approach”: it is  more  mathematical than  physical approach, but leads to the same results, separating the zero eigenvalues and their corresponding eigenvectors. Zero eigenvalues are the consequence of the singularity of the Stiffness matrix of an element. The number of zero eigenvalues is equal to the loss of rank of the matrix, or the number of rigid body shapes inherent to the element.

The Natural Factor formulation is advantageous in two different aspects: First, the algebraic properties of the Natural Factor, and the Householder transformation (transforming the natural factor to an upper triangular matrix) turn the process of  assembly of the overall matrix and  further corrections, into an elegant and simple algebraic chain of operations. Second, the Natural Factor calculation yields more accurate results, especially in ill-conditioned systems. In addition, numerical stability remain even while a given structure is close to become a mechanism.

The Natural Factor Method , is not able to describe the dynamic behavior of an element. The most important condition for  it existence is not fulfilled: internal forces due to rigid body motion must vanish. In a system given in dynamic loading, these forces do not vanish, because of the inertial forces.